In the given figure, line $PY\parallel $ side BC, $AP = 4$ , $PB = 8$ , $AY = 5$ and $YC = x$ . Find x.

Hint: We can use the concept of parallel lines intersected by a transversal to show that the corresponding angles of the triangles are equal. Then we can prove that the triangles are similar. Then we can take the sides as proportional. On substituting and solving for x we can obtain the required value of x.

Complete step-by-step answer:
It is given in the question that $PY\parallel BC$ ,
Consider AB as a transversal cutting the parallel lines PY and BC.
Then $\angle APY$ and $\angle ABC$ are corresponding angles. We know that when 2 parallel lines are intersected, their corresponding angles will be equal.
  $ \Rightarrow \angle APY = \angle ABC$ .. (1)
Now consider AC as a transversal cutting the parallel lines PY and BC.
Then $\angle AYP$ and $\angle ACB$ are corresponding angles.
  $ \Rightarrow \angle AYP = \angle ACB$ .. (2)
Now consider the triangles APY and ABC,
 $\angle YAP = \angle CAB$ as it represents the same angle.
From (1) and (2),
 $ \Rightarrow \angle APY = \angle ABC$
 $ \Rightarrow \angle AYP = \angle ACB$
As the 3 corresponding angles are equal, the triangles are similar.
 $ \Rightarrow \Delta APY \simeq \Delta ABC$
We know that if 2 triangles are similar, their sides will be proportional. So, we can write,
 $\dfrac{{AP}}{{AB}} = \dfrac{{AY}}{{AC}}$
From the figure, $AB = AP + PB$ and $AC = AY + YC$ .
 $ \Rightarrow \dfrac{{AP}}{{AP + PB}} = \dfrac{{AY}}{{AY + YC}}$
On substituting the values, we get,
 $ \Rightarrow \dfrac{4}{{4 + 8}} = \dfrac{5}{{5 + x}}$
 $ \Rightarrow \dfrac{4}{{12}} = \dfrac{5}{{5 + x}}$
On rearranging, we get,
 $ \Rightarrow 5 + x = \dfrac{{5 \times 12}}{4}$
 $ \Rightarrow x = 15 - 5$
 $ \Rightarrow x = 10$
Therefore the value of x is 10.

Note: We used the concept of parallel lines intersected by a transversal to prove the equality of the angles. Then we used the concept of similarity of triangles. Two triangles are said to be similar if all the angles are equal or all the sides are proportional. When we take the proportionality of the sides, we must take the sides of the triangle not a part of the side. While taking the proportionality, we must make sure that sides of the same triangle come in the numerators on both sides.